A family of triangular grids in digital geometry

In this paper we show a new geometric interpretation of the hexagonal and triangular grids. They can be considered as the sets of points of one (see (I. Her, 1995)), respectively two plane(s) in Z³. By this approach we can build up a whole family of triangular grids (the so called n-planes triangular grids). The hexagonal and triangular grids are the first two members of this family, moreover, they are duals of each other. We investigate the three-planes grid, the third member of the family, and its dual in detail. We show that for n ≥ 4 on, the n-planes triangular grids are non-planar.